CRACK OPENING STRETCH

IN A PLATE OF FINITE WIDTH

bY

F. ERDOGAN and €4, BAKIOGLU

Lehigh University, Bethlehem, Pa.

(HASA-CR-14C5E3) C R A C K O P Z N i N G STRETCB IN A PLATE CE FLNITE YIDTH (Lenigh Univ.) 2 1 p HC 34.25 C S C L 26K

September 1974

N74-35308

Urclas G3/32 52715

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

GRANT NGR 39 - 007-011

https://ntrs.nasa.gov/search.jsp?R=19740027195 2018-06-01T12:48:36+00:00Z

CRACK OPENING STRETCH 13 A PLATE OF FINITE WICTH(*)

F. Erdogan and M. Bakioglu Lehigh U n i v e r s i t y , Bethlehem, Pa.

Abstract

The problem of a u n i a x i a l l y s t r e s s e d p l a t e of f i n i t e w id th con ta in ing a c e n t r a l l y located "damage zone" i s cons idered . I t is assumed t h a t t h e f l aw may be r e p r e s e n t e d by a par t - through c r a c k pe rpend icu la r t o the p l a t e s u r f a c e , t h e n e t l igaments i n t h e p l a n e of t h e crack and throhgh- the- th ickness nar row s t r i p s ahead of t h e crack ends are f u l l y y i e l d e d , pnd i n t h e y i e l d e d s e c t i o n s t h e ma- t e r i a l may c a r r y on ly a c o n s t a n t normal t r a c t i o n w i t h magnitude q u a l t o the y i e l d s t r e n g t h . The problem is so lved by neglect ing t h e bending e f f e c t s and t h e crack opening s t r e t c h e s a t t h e c e n t e r and t h e ends of t h e c rack are ob ta ined . Some a p p l i c a t i o n s of t h e r e s u l t s are i n d i c a t e d by u s i n g t h e concepts of c r i t i ca l c rack open- i n g s t r e t c h and c o n s t a n t s l o p e p l a s t i c i n s t a b i l i t y .

1. INTRODUCTION

I n t h i s paper w e r e c o n s i d e r t h e problem of an i n f i n i t e s t r i p

or a long p l a t e of f i n i t e width w i t h a symmetr ica l ly l o c a t e d crack

pe rpend icu la r t o t h e boundar ies (F igu re 1). The e l a s t i c i t y problem

f o r t h i s geometry has been cons idered b e f o r e i n v a r i o u s p u b l i c a t i o n s

(e .g . , [ l -4) . I n a c o n f i g u r a t i o n such as t h e cracked s t r i p , u n l e s s

t h e material i s ext remely " b r i t t l e " ( i n t h e sense t h a t it may r u p t u r e

wi thou t undergoing a p p r e c i a b l e i n e l a s t j c deformat ions around t h e

crack t i p s ) , f o r c r ack l e n g t h s of t h e o r d e r of t h e width of n e t l i g -

aments it i s clear LIiat t h e e l a s t i c i t y s o l u t i o n may n o t be adequate

e i t h e r t o d e s c r i b e t h c stress and deformat ion s t a t e s i n t h e s t r i p o r

as a p r e d i c t i o n t o o l i n t h e r e l a t e d f r a c t u r e s tudy . On t h e o t h e r

hanA under "p lane stress" c o n d i t i o n s if the m a t e r i a l undergoes

p 'as t ic deformat ions w i t h t h e p l a s t i c zone s i z e around t h e c r a c k

(*)Tr,is work was suppor ted by NASA-Laugley under t h e G r a n t NGR39- 007-901 a3d by t h e Na t iona l Science Foundation under t h e Grant GK-42771X.

1

t i p s of t h e order of crack l e n g t h , there is a t least a c e r t a i n schoo l

of thought adher ing t o the n o t i o n t h a t t h e concepts of t h e crack

opening s t r e t c h or the p l a s t i c i n s t a b i l i t y may be used for f r a c t u r e

c h a r a c t e r i z a t i o n of t h e material.

Aside from the v e r i f i c a t i o n of t he v a l i d i r y of these concep t s

and t h e expe r imen ta l de t e rmina t ion of t h e related s t r e n g t h parameter

f o r a g iven material, i n a p p l i c a t i o n s one of t h e impor t an t problems

is, of c o u r s e , a n a l y t i c a l l y f i n d i n g a r easonab le estimate of the

crack opening stretch o r the v a l u e of the e x t e r n a l load a t p l a s t i c

i n s t a b i l i t y . Even though t h e r e is no widely accepted s t a n d a r d i z a -

t i o n , the s t r e n g t h parameter of t h e material, t h e so -ca l l ed cri t i-

cal crack opening stretch, is d e f i n e d as the relative d isp lacement

measured a t t h e crack t i p a t t h e o n s e t of r u p t u r e ( th rough , fo r

example, pho tograph ica l ly recorded diamond i n d e n t a t i o n marks) .

As fo r t h e q u a n t i t y r e p r e s e n t i n g t h e s e v e r i t y of t h e e x t e r n a l

loads, ever s i n c e t h e p u b l i c a t i o n of Lugdale 's work on t h e s u b j e c t

[SI, t h e c rack opening d isp lacement c a l c u l a t e d a t t h e c rack t i p by

u s i n g t h e conven t iona l p l a s t i c s t r i p model has been cons ide red t o

be q u i t e adequate .

I n t h i s paper w e w i l l mainly be in t e re s t ed i n s t u d y i n g t h e

e f f e c t of crack l e n g t h - t o - p l a t e w i d t h r a t i o , a/h on t h e p l a s t i c

zone s i z e and t h e crack opening d isp lacement , d i n p l a t e s of f i n i t e

width. I t w i l l be assumed t h a t t h e p l a t e has e i t h e r a s y m e t r i c a l l y

l o c a t e d through crack o r a "damage zone" which may be r e p r e s e n t e d

by a par t - through crack (F igu re 1) . I n cons ide r ing t h e par t - through

c rack problem the bending e f f e c t s w i l l be neg lec t ed and it w i l l

f u r t h e r be assumed t h a t t h e n e t l igaments connec t ing t h e par t - through

c rack t o t h e p l a t e s u r f a c e s a s w e l l as a narrow through t h e t h i c k -

2

ness s t r i p i n t h e p l ane a33 ,,head of t h e c rack is f u l l y y i e l d e d

(shaded area i n F igure 1 b ) . A f t e r c a l c u l a t i n g 6 as a f u n c t i o n

of a/h and t h e a p p l i e d load, one may estimate t h e load c a r r y i n g

c a p a c i t y of t h e p l a t e based on a c r i t e r i o n of c r i t i ca l c rack open-

i n g s t r e t c h or p l a s t i c i n s t a b i l i t y . Such estimates are also g i v e n

i n t h e paper.

2. FORMULATION OF THE PROBLEM

The problem i s solved by us ipg t h e s t anda rd s u p e r p o s i t i o n

technique and t h e method described i n [3 ] . I n [3] it w a s shown

t h a t the p lane elastostatics p rob len aJf an i n f i n i t z s t r i p -h<x<h,

-my<=, c0ntainir.c; a c rack a long -a<x<d, y=O may be formula ted i n

terms of t:Je f o l l o x i n g i n t e g r a l equat ion:

1+ K a 1 & [ A - L k ( x , t ) ] G ( t ) l t = - -r p ( x ) , ( -a<x<a) - t - x ' 4P

where

is t h e c rack s u r f a c e d isp lacement ( i n y - d i r e c t i o n ) ,

p ( x ) = -u (x,O), O=a (x,O) , (-a x a) (3) YY XY

are t h e c rack s u r f a c e t r a c t i o n s which are assumed t o be t h e o n l y

e x t e r n a l loads a c t i n g on t h e s t r i p , p and K ( K = 3-4u f o r p l ane s t r a i n ,

~ = ( 3 - v ) / ( l + v ) for t h e g e n e r a l i z e d p l ane stress, v being t h e P o i s s o n ' s

ra t io) are t h e e las t ic c o n s t a n t s , and t h e k e r n e l k ( x , t ) i s g i v e n by

d s I 03 - (h - t ) k . ( x , t ) = I, K ( x , t , s ) e

K (x , t , s ) = e-hs { - [ 1+ (3+2hs) e -2hs] cosh (xs)

- 2 x ~ e ~ ~ ~ ~ s i n h ( x s ) - [2xs s i n h ( x s )

+(3-2hs+e -2hs) cosh (xs ) I +4hse -2hs-,- 4hs)

[ l - 2 s ( h - t ) 1 ) / (1

( 4 )

3

The index of t h e s i n g u l a r i n t e g r a l equa t ion (1) i s +1, hence i t s

s o l u t i o n is determina te w i t h i n one a r b i t r a r y c o n s t a n t which is

determined from t h e f o l l o w i , , g sih e-valuedness c o n d i t i o n : 9l la G(x)dx=O -a (6)

Once t h e d e n s i t y f u n c t i o n G ( x ) i s determined a l l t h e d e s i r e d f i e l d

q u a n t i t i e s i n t h e problem may be expressed i n terms of d e f i n i t e i n -

t e g r a l s w i t h G as t h e d e n s i t y and t h e related Green ' s f u n c t i o n s as

t h e ke rne l s .

R f e r r i n g t o F i g u r e 1 l e t 2h be t h e width of t h e p l a t e , 2a and

b t h e "equiva len t" dimensions of t h e par t - throuch c rack r e p r e s e n t i n g

t h e i n i t i a l damage zone, bo t h e p l a t e t h i c k n e s s , a - a t h e s i z e of

t h e p l a s t i c zone i n t h e p l ane of t h e c rack , and CJ t h e uniform ten-

sile stress a c t i n g on t h e p l a t e away from t h e c rack r eg ion . It w i l l

be assumed t h a t i n a d d i t i o n t o t h e "narrow s t r i p s " of l e n g t h a - a P i n t h e p l ane and ahead of t h e i n t e r n a l c r ack , t h e n e t l igaments

connect ing t h e c rack t o t h e p l a t e s u r f a c e s are also f u l l y y i e l d e d .

Thus, t h e y i e l d e d zones are shown i n 7 i g u r e 1 a s t h e shaded r e g i o n s .

I n formula t ing t h e problem i t w i l l be assumed t h a t t h e damage zone

can be approximated by a r e c t a n g u l a r par t - through crack of dirnen-

s i o n s 2a and b wi th s i d e s p a r a l l e l and pe rpend icu la r t o t h e p l a t e

s u r f a c e s , and t h e bending e f f e c t s a r i s i n g from t h e nonsymmetric

o r i e n t a t i o n and shape of t h e par t - throuqh crack i s n e g l i g i b l e . The

s o l u t i o n of t h e problem may then be ob ta ined by t h e s u p e r p o s i t i o n

of s o l u t i o n s of t h e fo l lowing t h r e e problems:

Problem A : No crack: e x t e r n a l l o a d :

P

0

4

Problem B: Crack : -a <x<ap, y=O; P

e x t e r n a l l oad : u ( x , O ) =-p (x) =-uo, (-a <x<a ) : YY P P

Problem C : C r a c k : -a <x<a y=O; P P'

e x t e r n a l load : u (x,O) =-p (x) =uy YY

b -b (x ,0)=-p(x)=uyl l ; -

b0 P' aYY

fo r a< Cxa< a

f o r -a<x<a.

Here t h e dlmensions bo, h , b, a (F igure 11, t he e x t e r n a l l oad u,-,

and t h e y i e l d s t r e n g t h u are known. The f i c t i t i o u s c rack length Y 2a is unknown and i s determined from t h e c o n d i t i o n of f i n i t e n e s s

of stress state a t the crack t i p s ;a . P

P Note that t h e problem h a s a symmetry i n load ing and geometry

w i t h r e s p e c t t o y axis and t h e Problem A has no c o n t r i b u t i o n to

the stress s i n g u l a r i t i e s . The re fo re , t h e c o n d i t i o n g i v i n g a may P

be expressed as

kB+kC=O (7)

where kB and kC a r e t h e stress i n t e n s i t y f a c t o r s a t a o b t a i n e d

from t h e s o l u t i o n s of Problems B and C , r e s p e c t i v e l y and may be

expressed i n terms of t h e unknown d e n s i t y f u n c t i o n G(x) as f o l l o w s :

P

k =- '' l i m J 2 ( a n ) G j ( x ) , (j=B,C). j 1 + K x-ta P

P If w e d e f i n e t h e fo l lowing d imens ionless parameters

X=a/h, X =a /h, ( 9 ) P P equasion (7 ) may be w r i t t e n as

The problem i s so lved simply i n an i n v e r s e manner, i .e . , f o r a f i x e d

, X i s v a r i e d , KB and KC are found ( f o r u n i t l oads ) from (1) and XP (81 , t h e corresponding load r a t i o ao/u i s found from (lo), and t h e n

Y

5

t h e curves g i v i n g (a/a ) = ( X / X ) v s ao/a

a parameter.

are prepared w i t h X a s P P Y P

A f t e r de te rmining t h e d e n s i t y f u n c t i o n

G (x) =GB (XI +Gc (x)

v ( x , + O ) = -kaPG (x) dx. ( 1 2 )

(11)

The c rack s u r f a c e d isp lacement may be e v a l u a t e d from

From (12) the crack opening stretches of p h y s i c a l i n t e r e s t , i.e.8

t h a t a t t h e a c t u a l c r ack t i p x=a, and a t t h e c e n t e r x=O may be

ob ta ined as 2

6=v (a,+O) -v (at-0) = -21, G(x)dx , (13)

d O ~ ( O , + O ) - ~ ( O , - O ) = -2I:PG(x)dx. (14)

3. THE IESJLTS

The c a l c u l a t e d r e s u l t s which are shown i n F i g u r e s 2-11 are ob-

t a i n e d f o r t h r e e v a l u e s of r e l a t i v e area of t h e par t - through c rack ,

namely (b/bo)=0.5, 0 .75, and 1 ( i . e . , th rough c r a c k ) . F i g u r e s 2-4

g i v e t h e in fo rma t ion t o o b t a i n t h e p l a s t i c zone s i z e o r a fo r a

g iven load r a t i o ao/a

p lane is f u l l y y i e l d e d and w e have

P and crack l e n g t h , a. F o r h =1 t h e crack

Y P

g i v i n g t h e s t r a i g h t l i n e s shown i n the f i g u r e s . The v a l u e of

X = ( a /h ) = 0 cor responds t o t h e i n f i n i t e p l ane f o r which i n t h e

case of through crack (b=bo) w e have 'p i g u r e 4 ) P P

The crack opening s t re tch 6 c a l c u l a t e d a t t h e c rack t i p x=a

(see equa t ion 13) i s shown i n F i g u r e s 5-7. The no rma l i za t ion

6

f a c t o r d used i n t h e s e f i g u r e s i s g iven by

d=1+K a= 4 a ~ a (p l ane stress). - r y E I n t h e f i g u r e s h=a/h is used as t h e parameter . The v a l u e h=O aga in

corresponds to t h e i n f i n i t e p l ane f o r which i n t h e case of through

c rack w e have (F igu re 7)

- =log ( c o s - l r U 0 ) . a= lr 2a 6

Y (18)

The asymptot ic v a l u e s of t h e 6 curves i n d i c a t e d i n t h e f i g u r e s are

the load ratios corresponding t o t h e f u l l y - y i e l d e d n e t s e c t i o n and

are g iven by

(19) U b - 1- --A. Y b* U

N o t e t h a t i n l i m i t when X = l t h e (&/dl vs (ao/a ) curve reduce t o

t h e s t r a i g h t l i n e (ao/ay)= l-b/bo shown i n the f i g u r e s . Y

I n t h e case of a par t - through crack, from t h e view p o i n t of

a p p l i c a t i o n s a more impor tan t q u a n t i t y i s t h e c rack opening stretch

6 o c a l c u l a t e d a t x=O, i .e. , t h e maximum s t r e t c n .

show t h i s q u a n t i t y f o r (b/bo)= 0 . 5 and 0 . 7 5 , r e s p e c t i v e l y .

asymptot ic v a l u e s of these 6 o curves too are g iven by ( 1 9 ) .

F i g u r e s 8 and 9

The

I f one adopts a c r i t i c a l c rack opening stretch c r i t e r i o n f o r

r u p t u r e t h e n t h e load-car ry ing c a p a c i t y of t h e p l a t e may be o b t a i n e d

from

6msx- - 6 cr . (20)

where 6max i s t h e measure of t h e i n t e n s i t y or' t he a p p l i e d l o a d wi th

6max=60 F i g u r e 1 0 shows t h e load-car ry ing c a p a c i t y of t h e p l a t e wi th a

through crack f o r v a l u e s of c r i t i c a l c r ack opening s t re tch

f o r t h e pa r t - th rough c rack and 6max=6 for t h e through c rack .

0.1 2 (6 /d) Note t h a t f o r c o n s t a n t 6 t h e d e r i v a t i v e of cr

7

u vs A=a/h curve is n e g a t i v e , meaning t h a t , accord ing t o t h i s

c r i t e r i o n t h e r u p t u r e p rocess i s u n s t a b l e . F i g u r e 1 0 i s ob ta ined

from t h e c a l c u l a t e d r e s u l t s g i v i n g F i g u r e 7. F o r a p l a t e w i t h a

par t - through crack s imi l a r curves g i v i n g t h e load-car ry ing c a p a c i t y

may be ob ta ined f r o m F i g u r e s 8 and 9. The s t r a i g h t A n e shown i n

F i g u r e 1 0 corresponds t o t h e f u l l y - y i e l d e d n e t s e c t i o n . I n t h i s

case, s i n c e t h e material i s assumed t o have no s t r a in -ha rden ing ,

the corresponding v a l u e of t he c rack opening s t r e t c n is v e r y l a r g e

( t h e o r e t i c a l l y , i n f i n i t e ) .

0

I t should be po in ted o u t t h a t the r e s u l t s found i n t h i s paper

f o r plates w i t h f i n i t e w i d t h are s imilar t o t h o s e found i n s h e l l s ,

A=a/h p l ay ing t h e role of t h e s h e l l parameter X=[12(1-u ) ] 2 1 / 4

*(a/%) (wi th R t h e r a d i u s of c u r v a t u r e , bo t h e t h i c k n e s s ) .

(see for example [ 6 , 7 ] ) . I n s tudy ing t h e b u r s t phenomenon i n

c y l i n d r i c a l s h e l l s , i t was observed t h a t one may a lso use t h e v a l u e

of 6 corresponding t o a s t a n d a r d s l o p e i n 6/d v s ao/oy p lo t ra ther

than a f i x e d va lue of 6 i t s e l f as t h e r e p r e s e n t a t i v e of t h e i n t e n -

s i t y of t h e a p p l i e d l o a d s [ 8 ] . The r a t i o n a l here be ing t h a t , as

seen f r o m F i g u r e s 5 - 8 , a f t e r reaching a c e r t a i n v a l u e any f u r t h e r

i n c r e a s e i n t h e a p p l i e d load may cause v e r y l a r g e i n c r e a s e s i n t h e

c rack opening stretch which may be i n t e r p r e t e d as t h e o n s e t of

"necking" p rocess , hence p l a s t i c i n s t a b i l i t y . F i g u r e 11 shows t h e

load-car ry ing c a p a c i t y of a p l a t e w i th a through c rack ob ta ined by

us ing t h i s concept for v a r i o u s s t anda rd s l o p e s .

F i n a l l y it should be emphasized t h a t i n t h e model used i n t h i s

paper t h e e f f e c t of t h e s t r a i n hardening h a s n o t been t aken i n t o

account. Hence, i n a p p l i c a t i o n s ay of t h i s paper should be con-

8

s i d e r e d as a "flow stress" rather than t h e s t a n d a r d 0.2 p e r c e n t

o f f s e t y i e l d s t r e n g t h qf t h e material. The v a l u e of t h e flow

stress a may be s e l e c t e d as e i t h e r :=(a

where u

and a a f ixed parameter , O<a<(uU - u y ) / a y .

u ) / z or a= (1 + a ) u y Y + u is t h e s t a n d a r d y i e l d s t r e n g t h ; uu t h e u l t i m a t e s t r e n g t h Y

Bferences :

1. M. I s i d a , "Crack t i p stress i n t e n s i t y f a c t o r s f o r t h e t e n s i o n of an e x c e n t r i c a l l y craiked s t r i p " , J. Appl. Mech.,Vol. 33, Trans. ASME, p. 674 (191 .

2. I. N. Sneddon and F. P. S r i v a s t a v , "The stress f i e l d i n t h e v i c i n i t y of a G r i f f i t h Crack i n a s t r i p of f i n i t e wid th" , I n t . J. Engng. S c i . V o l . 9 , p. 479 (1971) .

i n f i n i t e strip'*, J. Appl. Mech. 1 7 0 1 . 42 (1975) ( t o a p p e a r ) . 3. G.D. Grupta a n d F . Erdogan, "The problem of edge c racks i n an

4. S. Krenk and M Bakioglu, "Transverse c r a c k s i n a s t r i p w i t h r e i n f o r c e d s u r f a c e s " , I n t . J. of F r a c t u r e , Vol. 11, (1975) ( t o appea r ) .

5. D.S. Ibgdale , "Yielding of steel s h e e t s con ta in ing slits", J. Mech. Phys. S o l i d s , Vo l . 8 , p. 100 (1960) .

6. 7 . Erdogan and M. kitwani , " P l a s t i c i t y and t h e c rack opening d,;placement i n s h e l l s " , I n t . J. of F r a c t u r e Mechanics, 701. 8, p. 413 (1972).

7. P. Erdogan and M. Patwani, ' ' F rac tu re i n i t i a t i o n and propoga- t i o n i n a c y l i n d r i c a l s h e l l c o n t a i n i n g an i n i t i a l s u r f a c e flaw", Nuclear Engineer ing and Cesigr . ,Vol . 27, p . 1 4 (1974) .

8. F. Erdogan and M. a t w a n i , "The u s e of COf: and p l a s t i c i n s t a b i l i t y i n c rack propogat ion and a r r e s t i n she l l s " , Proceedings of t h e Symposium on "Crack 'ropogation i n P i p e l i n e s " , Newcastle upon Tyne, England !March, 1974) .

9

Fig. 1. S t r i p and crack geometry.

0 II

I1 Q

x

J L n

0 0 II

Q 0

al C

P a, a 0 cl

a a,

$ a,

0 c, rl)

a

3 u .

(v

tn -4 Irr

a, N

.rl v)

a? 0

a, c 0 N

W 0'

Q 0

v)

0 k

u) b

0 II

\ a 0 -4 4J m k

E 4J a a, a x 0 m k u k 0

W

0 n

- It

9 cu 0

0

Q1 N 4 m

. X u a k 0

c P 1 0 k c 4J LI 0 W

0.8

0.6

0.2

'2.

0 0.2 0.4 0.6 0.8 I .o Qa uY 7

Fig. 5 . The crack opening stretch 6 at the crack t i p f o r b/bo=O

(d=40y a/E) . . 5

‘7

(d=4a, a/E) .

. 1.0 -

I

. 0.8 -

. - 8 - d

- 0.6 -

- - -

I I

0 0.2 0.4 0.6 0.8 I .o

*

F i g . 7 . The crack openiiig s t r e t c h 6 at the crack t i p f o r through

crack (d=4ay a/C).

b - = 0.5 b0

0.3,

0 0.2 0.4 0.6 0.8 I .o - CO

Fia. 8. The crack opening s t r e t c h 6 o a t t h e c e n t e r x=O for b/bo=O. 5 (d=4uy a/E) .

I .o

0.8

0.6

- 80 d

0.4

0.2

0

I I

I

I

0.2 0.4 0.6 0.8 I .o =0

=Y -

F i g . 9. The crack opening s t r e t c h 6o a t t h e c e n t e r x=O for

b/bo=0.75 (d=40y a/E) .

c r i t i c a l c r a c k opening s t r e t c h c r i t e r i o n (A=a/h).

1.0 Slope:

I I I I I I I I I I

0 0.5 I .o x

F i g . 1 1 . Load c a r r y i n g c a p a c i t y of a c racked p i a t e based on

c o n s t a n t s lope - p l a s t i c i n s t a b i l i t y (A= a/h).